In recent years, there has been an explosive increase in the amount of digital image data. The requirements for its storage. and communication can be reduced considerably by compressing the data while maintaining their visual quality. The work in this thesis is concerned with the compression of still images using fixed and adaptive wavelet transforms. The wavelet transform is a suitable candidate for representing an image in a compression system, due to its being an efficient representation, having an inherent multiresolution nature, and possessing a self-similar structure which lends itself to efficient quantization strategies using zerotrees. The properties of wavelet transforms are studied from a compression viewpoint. A novel augmented zerotree wavelet image coding algorithm is presented whose compression performance is comparable to the best wavelet coding results published to date. It is demonstrated that a wavelet image coder performs much better on images consisting of smooth regions than on relatively complex images. The need thus arises to explore the wavelet bases whose time-frequency tiling is adapted to a given signal, in such a way that the resulting waveforms resemble closely those present in the signal and consequently result in a sparse representation, suitable for compression purposes. Various issues related to a generalized wavelet basis adapted to the signal or image contents, the so-called best wavelet packet basis, and its selection are addressed. A new method for wavelet packet basis selection is presented, which aims to unite the basis selection process with quantization strategy to achieve better compression performance. A general zerotree structure for any arbitrary wavelet packet basis, termed the compatible zerotree structure, is presented. The new basis selection method is applied to compatible zerotree quantization to obtain a progressive wavelet packet coder, which shows significant coding gains over its wavelet counterpart on test images of diverse nature.